At first glance, you might be curious as to what this post is about. In that case, I would recommend learning the difference between ‘fallacies’ and ‘phalluses’. Dirty buggers. No, today won’t be spent talking about the male appendage – rather, some maths! Don’t sound too excited…

Yes, this post will be all about mathematical fallacies: these are ‘mistakes’ in mathematical proofs. However, the difference between a genuine mistake in a proof and a *fallacy* is that a fallacy will often lead to absurd results and conclusions that may seem flawless. Only when we examine the proof itself do we find the fallacy. So, today I’m going to show you some famous-ish fallacies that I find particularly interesting.

**1. Fun With Fractions**

Fractions is probably the part of maths that most people hate. Personally, I don’t mind them, but to some people they are the spawn of Satan. After all, there are so many rules to follow: *“when can I divide?”; “Which fraction do I flip to divide?”; “Why am I even doing this?!”* Many many questions to ponder. Below is a little example of some of the misunderstandings that go on when people deal with fractions. Now be warned, this is NOT how you simplify fractions. It just so happens that it works for this one example. Try it: type into your calculator ’16/64′ and it’ll simplify to 1/4. To all you non-mathematicians, sit back and marvel at how delightfully simple this fallacy is; to all you mathematicians, I challenge you to find another example where this kind of error works.

**2. Imaginary Numbers**

To many people, the thought of imaginary numbers is ridiculous. In fact, I would wager that a lot of people wish that *all *numbers were imaginary and thus irrelevant. But I’m afraid that *these *imaginary numbers are far from irrelevant. The main thing to be aware of in terms of imaginary numbers is that the square root of -1 is called ‘*i*‘. Now, using that knowledge, behold as I prove that 1 = -1.

At first glance this proof seems completely logical, however, there is an ever-so-tiny thing wrong with it. Now I could explain why this proof makes no sense, but it’s far more fun to let you work it out for yourselves. So, if you think you know the answer, please leave a comment!

**3. MORE Imaginary Numbers**

Wow, people just cannot get enough imaginary numbers, eh? This little fallacy is in the same style as the previous one. This time, the proof is showing how the square root of -1 is just 1 (which we know is impossible – remember I said it was that funny little number called ‘*i*‘). So, have a bash at this one!

**Handy Hint: consider the fourth roots of 1…**

**4. Tricky trigonometry**

*‘SOHCAHTOA!’ – *No, that’s not a made-up Japanese word, but a way of remembering the 3 trigonometric formulas! (‘Sine, Opposite, Hypotenuse, Cosine, Adjacent, Hypotenuse, Tangent, Opposite, Adjacent’) Ah, such fun! Trigonometry is the study of triangles and the relationships between their angles and sides. You can do lots with good ol’ trig: calculate angles, calculate sides, erm…calculate…other things too. Anyway, this fallacy proves that 0 = 2! Doesn’t that sound dramatic? Well, you’re probably aware of the fallacious proof that 1 = 2 (done so by dividing by zero – a huuuuuge no-no in maths) but this proof uses trigonometry! Have a look-see at this bad boy!

**Handy Hint: it’s very close to number 3 in the list…**

So, I’m afraid our mathematical journey must come to an end! We’ve had some ups, some downs, but most importantly – we’ve learnt some maths. I appreciate that this entire post may have gone over some people’s heads, but hey, sometimes I like to do maths.

Till next time,

Tom.

The problem in the last example I found relatively quickly, you have to add the plus/minus sign when applying the square root. This takes care of it. But I can not figure out the second.example. The first two steps seem fine. Is it the third step? Can we apply the exponentiation identities here or do they only hold true for real numbers? ,,, Great post!

You’re absolutely right about the last example: we need to consider the negative square root. I can give you a little hint as to the problem with the second example: does separating sqrt(ab) into sqrt(a).sqrt(b) hold for all a,b? I’ll leave you with that one!

Hey Tom, thanks for the tip, I finally figured it out. The rule sqrt(ab) = sqrt(a) * sqrt(b) only holds true for positive a,b. Fantastic example, thanks a lot!